Download strategy for integration

Ch8: STRATEGY FOR INTEGRATION
integration is more challenging than differentiation.
No hard and fast rules can be given as to which
method applies in a given situation, but we give
some advice on strategy that you may find useful.
how to attack a given integral, you might try
the following four-step strategy.
Ch8: STRATEGY FOR INTEGRATION
4-step strategy
1
Simplify the Integrand if Possible
2
Look for an Obvious Substitution
function and its derivative
3
Classify the Integrand According to Its Form
Trig fns, rational fns, by parts, radicals,
4
Try Again
1)Try subsitution 2)Try parts 3)Manipulate integrand
4)Relate to previous Problems 5)Use several methods
Ch8: STRATEGY FOR INTEGRATION
4-step strategy
1
Simplify the Integrand if Possible
Ch8: STRATEGY FOR INTEGRATION
4-step strategy
2
Look for an Obvious Substitution
function and its derivative
 sin xdx
2

sec
xdx

  cos xdx
  csc
2
xdx
 sec x tan xdx
  csc x cot xdx
  e dx
x


dx
x
dx
x 2 1
Ch8: STRATEGY FOR INTEGRATION
4-step strategy
3
Classify the integrand according to Its form
Trig fns, rational fns, by parts, radicals,
8.2
4
8.4
8.1
8.3
Try Again
1)Try subsitution 2)Try parts 3)Manipulate integrand
4)Relate to previous Problems 5)Use several methods
Ch8: STRATEGY FOR INTEGRATION
Ch8: STRATEGY FOR INTEGRATION
3
1
Classify the integrand according to Its form
Integrand contains:
2
ln x
ln and its derivative
by parts
Integrand contains:
tan 1 , sin 1 
f and its derivative
by parts
4
Integrand radicals:
a2  x2 , x2  a2
3
8.3
by parts (many times)
poly
5
 sin
Integrand = 
f ( x) g ( x)
n
 (x
Integrand contains: only trig
x cos xdx
 f (sin x) cos xdx
 f (tan x) sec xdx
 sec x tan xdx
m
n
2
m
6
We know how to
integrate all the way
4
 3 x)e3 x dx
Integrand = rational
PartFrac
f & f’
8.2
7
Back to original 2-times by part  original
x
 e sin xdx  e cos xdx
x
8
Combination:  sec
3
xdx
Ch8: STRATEGY FOR INTEGRATION
 x( x  1)e dx
 x tan xdx 
x
1
5
sin x
dx
8
x
 cos
x
dx
1
3
102
dx
(4  x 2 )3 / 2
x5  2
 x 2  1 dx
sin 2 x
 1  cos 4 x dx
cos x
122
x 2 dx
dx
 9  sin 2 x  (4  x 2 )3/ 2
x
 e sin 2 xdx  x  22 dx
( x  1)( x  1)
dx
6
3
 x 1  x  csc 2 x cot 2 xdx
Partial fraction
 sin( 3x) cos(2 x)dx
111
 (2  tan x)
x
2
112
2
x
dx
sin( 2 x)dx

dx
x 2 4
cos x  cos 3 x dx
 (sin( 2 x)  2 cos x)e
sin x
dx
Subsit or combination
Trig fns
dx
 x  6  x 1
sin x
 csc 2 x  1 dx
x
3
1 4x2
 sin(ln
15dx
 2x2  x  2
radicals
by parts

x 2 dx
x 2 )dx
Ch8: STRATEGY FOR INTEGRATION
122
102
Partial fraction
x5  2
 x 2  1 dx

dx
3
x 1
111
112
Subsit or combination
x2
 ( x  1)( x 2  1) dx
 (sin( 2 x)  2 cos x)e

15dx
 x3  2 x 2  x  2
cos x
9  sin x
2
dx
sin x
radicals
dx
 (4  x 2 )3 / 2
dx

dx
x 1 x
x
Trig fns
sin 5 x
 cos8 x dx
 csc
6
sin 2 x
 1  cos 4 x dx
3
2 x cot 2 xdx
 (2  tan x) dx
2

cos x  cos 3 x dx
 sin( 3x) cos(2 x)dx
sin x
 csc 2 x  1 dx
dx
x 4
2
by parts
 x( x  1)e dx  x
 x tan xdx
x
1
e
x
sin 2 xdx
 sin(ln
x 2 )dx
2
x 2 dx
 (4  x 2 )3 / 2
sin( 2 x)dx

x 2 dx
1 4x
2

dx
x  6  x 1
Ch8: STRATEGY FOR INTEGRATION
Partial fraction
Subsit or combination
Trig fns
radicals
by parts
Ch8: STRATEGY FOR INTEGRATION
Partial fraction
Trig fns
Subsit or combination
by parts
radicals
Ch8: STRATEGY FOR INTEGRATION
Partial fraction
Trig fns
Subsit or combination
by parts
(Substitution then  combination)
radicals
Ch8: STRATEGY FOR INTEGRATION
Partial fraction
Trig fns
Subsit or combination
by parts
(Substitution then  combination)
radicals
Ch8: STRATEGY FOR INTEGRATION
CAN WE INTEGRATE ALL
CONTINUOUS FUNCTIONS?
if
f (x )
YES or NO
Continuous.
Will our strategy for integration
enable us to find the integral of
every continuous function?
Anti-derivative
YES or NO
F (x )
e
x2
exist?
dx
Ch8: STRATEGY FOR INTEGRATION
elementary functions.
polynomials,
trigonometric
rational functions
inverse trigonometric
power functions
hyperbolic
Exponential functions
inverse hyperbolic
logarithmic functions
all functions that obtained from above by 5-operations
, , , , 
CAN WE INTEGRATE ALL CONTINUOUS FUNCTIONS?
Will our strategy for integration enable us to
find the integral of every continuous function?
e
x2
dx
NO
YES
Ch8: STRATEGY FOR INTEGRATION
elementary functions.
FACT:
polynomials,
trigonometric
rational functions
inverse trigonometric
power functions
hyperbolic
Exponential functions
inverse hyperbolic
logarithmic functions
all functions that obtained from above by 5-operations
If g(x) elementary
, , , , 
g’(x) elementary
NO:
If f(x) elementary
CAN WE INTEGRATE ALL
CONTINUOUS FUNCTIONS?
x
Will our strategy for integration
enable us to find the integral of
every continuous function?
F ( x)   f (t )dt
a
need not be an
elementary
Ch8: STRATEGY FOR INTEGRATION
CAN WE INTEGRATE ALL
CONTINUOUS FUNCTIONS?
Will our strategy for integration
enable us to find the integral of
every continuous function?
FACT:
NO:
If g(x) elementary
If f(x) elementary
g’(x) elementary
x
F ( x)   f (t )dt
a
need not be an
elementary
f ( x)  e
x2
has an antiderivative
x
F ( x)   e dt
a
t2
is not an elementary.
This means that no matter how hard we try, we will never succeed in
evaluating in terms of the functions we know.
In fact, the majority of elementary functions
don’t have elementary antiderivatives.